Graphing Inequalities in Real-World Scenarios Challenge

A farmer has 100 meters of fencing to create a rectangular pen for his sheep. If the length of the pen is represented by x and the width by y, write the equations for the perimeter and graph the inequality representing the area that can be enclosed. What are the possible dimensions of the pen?

• The possible dimensions of the pen are 0 < x < 25 and 0 < y < 25, with maximum area at 12.5m by 12.5m.

• The possible dimensions of the pen are 0 < x < 100 and 0 < y < 100.

• The possible dimensions of the pen are 50 < x < 100 and 50 < y < 100.

• The possible dimensions of the pen are 0 < x < 50 and 0 < y < 50, with maximum area at 25m by 25m.

A school is planning a field trip and has a budget of $500. The cost per student is $20 for transportation and $15 for admission. Write the system of inequalities to represent the number of students that can attend. Graph the inequalities and determine the maximum number of students that can go on the trip.

• 25

• 20

• 14

• 10

A company produces two types of gadgets, A and B. Each gadget A requires 2 hours of labor and each gadget B requires 3 hours. The company has a maximum of 12 hours of labor available. Write the inequality representing the labor constraint and graph it. How many of each type of gadget can be produced?

• The company can produce only 4 gadgets A and 4 gadgets B.

• The company can produce up to 8 gadgets A and 2 gadgets B.

• The company can produce up to 6 gadgets A and 4 gadgets B, or any combination that satisfies 2x + 3y ≤ 12.

• The labor constraint is represented by 2x + 2y ≤ 12.

A local gym offers two types of memberships: a basic membership for $30 per month and a premium membership for $50 per month. If a family wants to spend no more than $200 per month on memberships, write the inequality and graph it. How many of each type of membership can they purchase?

• The family can purchase up to 6 basic memberships, 4 premium memberships, or any combination of both that satisfies 30x + 50y ≤ 200.

• The family can purchase up to 8 basic memberships.

• The family can spend $250 on memberships.

• The family can purchase 3 premium memberships and 2 basic memberships.

A pizza shop sells two sizes of pizzas: small and large. A small pizza costs $8 and a large pizza costs $12. If a customer has $60 to spend, write the inequality representing the total cost and graph it. What combinations of small and large pizzas can the customer buy?

• 8x + 12y ≥ 60

• The customer can buy combinations of small and large pizzas such that 8x + 12y ≤ 60, where x ≥ 0 and y ≥ 0.

• x + y ≤ 5

• 8x + 12y = 60

A concert venue has a seating capacity of 500. If tickets are sold for $20 each for general admission and $30 each for VIP seating, write the system of inequalities representing the ticket sales if the venue wants to make at least $10,000. Graph the inequalities and find the possible combinations of tickets sold.

• x + y >= 500 and 20x + 30y <= 10000

• x + y <= 400 and 20x + 30y >= 8000

• The system of inequalities is: x + y <= 500 and 20x + 30y >= 10000.

• x + y <= 600 and 20x + 30y >= 12000

A charity event is selling two types of tickets: regular tickets for $25 and VIP tickets for $50. If they want to raise at least $1,000, write the inequality and graph it. How many of each type of ticket must they sell to meet their goal?

• They must sell combinations of regular and VIP tickets such that 25x + 50y ≥ 1000, e.g., 0 regular and 20 VIP, or 10 regular and 10 VIP.

• They must sell 40 regular tickets.

• They must sell 10 VIP tickets and 5 regular tickets.

• They must sell 30 VIP tickets.

A bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 3 eggs and each vanilla cake requires 2 eggs. If the bakery has 30 eggs, write the inequality for the number of cakes that can be made and graph it. What combinations of chocolate and vanilla cakes can be baked?

• x + y ≤ 15

• x + 2y ≤ 30

• The combinations of chocolate and vanilla cakes that can be baked are all pairs (x, y) such that 3x + 2y ≤ 30, where x ≥ 0 and y ≥ 0.

• 3x + 2y ≥ 30

A clothing store has a sale on shirts and pants. Shirts are $15 each and pants are $25 each. If a customer has $100 to spend, write the inequality representing the total cost and graph it. What combinations of shirts and pants can the customer buy?

• (2 shirts, 4 pants)

• The customer can buy combinations of shirts and pants such as (0 shirts, 4 pants), (1 shirt, 3 pants), (2 shirts, 2 pants), (3 shirts, 1 pant), and (4 shirts, 0 pants).

• (0 shirts, 5 pants)

• (5 shirts, 0 pants)

A local farmer's market sells fruits and vegetables. Apples cost $2 per pound and carrots cost $1 per pound. If a customer wants to spend no more than $30, write the inequality and graph it. How many pounds of apples and carrots can the customer buy?

• The customer can buy 10 pounds of carrots and 5 pounds of apples.

• The customer can buy 25 pounds of apples and no carrots.

• The customer can buy up to 15 pounds of carrots or a combination of apples and carrots that satisfies 2x + y ≤ 30.

• The customer can buy 20 pounds of apples only.